problem-solver

Problem Solver

Structured problem-solving with step-by-step explanations.

Problem-Solving Framework

flowchart LR
    A[Understand] --> B[Plan]
    B --> C[Execute]
    C --> D[Verify]
    D -->|Error| B
    D -->|Correct| E[Explain]

---

Step 1: Understand (Identify)

Questions to Ask

1. What is given? List all known information
2. What is asked? Clearly state what to find
3. What type of problem? Categorize (algebra, kinematics, etc.)
4. What are the constraints? Any restrictions or conditions

Template

**Given:**
- [List all known values with units]

**Find:**
- [What we need to calculate]

**Type:**
- [Problem category]

---

Step 2: Plan (Strategy)

Choose Approach

Problem Type	Common Strategies
Equation solving	Isolation, substitution, factoring
Word problems	Define variables, write equations
Physics	Draw diagram, identify forces/energies
Proofs	Direct, contradiction, induction
Optimization	Take derivative, find critical points

Write Steps (Before Executing)

1. First, I will...
2. Then, I will...
3. Finally, I will...

---

Step 3: Execute (Solve)

Show All Work

**Step 1:** [Action]
[Mathematical work]
= [Result]

**Step 2:** [Action]
[Mathematical work]
= [Result]

**Step 3:** [Final calculation]
[Work]
= [Final answer with units]

Best Practices

• One step at a time
• Include units throughout
• Box or highlight final answer
• Keep work organized vertically

---

Step 4: Verify (Check)

Verification Methods

Method	How to Apply
Plug back in	Substitute answer into original equation
Units analysis	Check dimensions match expected result
Estimate	Is the magnitude reasonable?
Alternate method	Solve differently, compare results
Edge cases	Does answer work at extremes?

Template

**Check:**
- Substituting x = [answer] into original equation:
  [show substitution] = [expected value] ✓
- Units: [unit analysis]
- Reasonableness: [is the answer logical?]

---

Teaching Modes

Hint Mode

Give progressive hints, not full solutions:

Hint 1: "What type of problem is this?" (Identification) Hint 2: "What formula relates these quantities?" (Direction) Hint 3: "Try isolating variable X first." (Specific guidance)

Socratic Mode

Guide through questions:

• "What do you notice about the given information?"
• "What principle applies here?"
• "What would happen if you tried...?"

Full Solution Mode

Complete step-by-step solution with explanations.

---

Common Problem Patterns

Pattern 1: Solve for X

1. Simplify both sides
2. Collect X terms on one side
3. Isolate X

Pattern 2: Word Problem

1. Define variables
2. Translate words to equations
3. Solve system of equations
4. Answer in context

Pattern 3: Rate Problem

1. Identify rate (units per time)
2. Use: Amount = Rate × Time
3. Set up equation and solve

Pattern 4: Optimization

1. Write objective function
2. Take derivative
3. Set equal to zero
4. Solve for critical points
5. Test for max/min

---

Answer Template

## Problem
[State the problem]

## Solution

### Understanding
- **Given:** [knowns]
- **Find:** [unknown]

### Approach
[Brief strategy explanation]

### Work
**Step 1:** [action]
[calculation]

**Step 2:** [action]
[calculation]

### Answer
**[Final answer with units]**

### Verification
[Check work]